What is a cascaded convolutional neural network?
The paper you are citing is the paper that introduced the cascaded convolution neural network. In fact, in this paper, the authors say


  To realize 3DDFA, we propose to combine two achievements in recent years, namely, Cascaded Regression and the Convolutional Neural Network (CNN). This combination requires the introduction of a new input feature which fulfills the "cascade manner" and "convolution manner" simultaneously (see Sec. 3.2) and a new cost function which can model the priority of 3DMM parameters (see Sec. 3.4)


where 3DDFA stands for 3D Dense Face Alignment, the framework proposed in this paper for face alignment, in which a dense 3D Morphable Model (3DMM) is fitted to the image via cascaded CNNs (the regressor), where the term dense refers to the number of points of the face that will be modeled. See figure 1 of this paper, which should provide some intuition behind the purpose of this framework.

In section 3 (page 3), they also say


  In this section, we introduce how to combine Cascaded Regression and CNNs to realize 3DDFA. By applying a CNN as the regressor in Eqn. 1, Cascaded CNN can be formulated as:


\begin{align}
\mathbf{p}^{k+1} = \mathbf{p}^{k} + \text{Net}^{k}
(\text{Fea}(\mathbf{I}, \mathbf{p}^k)) \tag{1}\label{1}
\end{align}

where 


$k$ is the iteration number
$\mathbf{p}$ is the regression objective 
$\text{Net}$ is the CNN structure 
$\text{Fea}$ contains the two constructed image features 


Pose Adaptive Feature (PAF) (section 3.2.1)
Projected Normalized Coordinate Code (PNCC) (section 3.2.2)

$\mathbf{I}$ is the image


The expression cascaded CNN apparently refers to the fact that equation \ref{1} is used iteratively, so there will be multiple CNNs, one for each iteration $k$. In fact, in the paper, they say


  Unlike existing CNN methods that apply different network structures for different fitting stages, 3DDFA employs a unified network structure across the cascade. In general, at iteration $k$ ($k = 0, 1, \dots, K$), given an initial parameter $\mathbf{p}^k$, we construct PNCC and PAF with $\mathbf{p}^k$ and train a two-stream CNN $\text{Net}^k$ to conduct fitting. The output features from two streams are merged to predict the parameter update $\Delta \mathbf{p}^k$
  
  $$
\Delta \mathbf{p}^k = \text{Net}^k(\text{PAF}(\mathbf{p}^k, \mathbf{I}), \text{PNCC}(\mathbf{p}^k, \mathbf{I}))
$$
  
  Afterwards, a better intermediate parameter $\mathbf{p}^{k+1} = \mathbf{p}^k + \Delta \mathbf{p}^k$ becomes the input of the next network $\text{Net}^k$ which has the same structure but different weights with $\text{Net}^k$.


In figure 2 of the paper (page 4), the structure of this two-stream CNN, $\text{Net}^k$, at iteration $k$, is shown.